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刚度较弱是椭球腔的一个弱势,因此,由氦压波动、机械振动及洛伦兹力等产生的失谐分析是研制椭球腔的必要环节。采用COMSOL Multiphysics[5]软件和CST[6]软件对氦压波动和洛伦兹力失谐进行分析,对于CSNS-II 的648 MHz 5-cell椭球纯铌腔,腔体所用铌材的机械特性如表1所列。
表 1 铌材特性
参量 值 杨氏模量/GPa 123 泊松比 0.38 密度/(kg/m3) 8.57×103 壁厚/mm 3.8 -
氦压的波动会引起超导椭球腔的形状变化,进而引起腔体本征频率的漂移。本征频率的偏移量∆f与氦压的变化量∆P满足式(1)[7]:
$$\Delta f = {K_{\rm{P}}} \bullet \Delta P,$$ (1) 其中KP定义为氦压敏感性系数,用来表征超导椭球腔对氦压波动的响应。
在两端束管端口固定边界条件下,对没带加强环的CSNS-II椭球腔,利用软件COMSOL Multiphysics计算其在不同氦压偏移量下的本征频率偏移,拟合曲线如图3所示,KP为-45.705 Hz/mbar。
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洛伦兹力是由腔体内电磁场产生,作用于腔体的内表面上,导致腔体产生形变失谐,力的大小由式(2)计算得到[8]:
$$P = \frac{1}{4}({\mu _0}{H^2} - {\varepsilon _0}{E^2}),$$ (2) 其中: μ0是真空磁导率;ε0是真空介电常数;H和E分别是腔体内表面磁场强度和内表面电场强度,它们的大小与超导腔的加速梯度Eacc呈正比关系。
洛伦兹力导致腔体的形变失谐称为洛伦兹力失谐,洛伦兹力失谐量由式(3)给出:
$$\frac{{\Delta f}}{{{f_0}}} = \frac{1}{{4W}}\int_{\Delta V} {({\varepsilon _0}{E^2} - {\mu _0}{H^2})} {\rm{d}}V,$$ (3) 其中:
$$W = \frac{1}{4}\int_V {({\varepsilon _0}{E^2} + {\mu _0}{H^2})} {\rm{d}}V,$$ (4) W是超导腔内的高频储能,与腔的加速梯度Eacc有关,f0是超导腔原有的谐振频率。
洛伦兹力对椭球腔的作用效果如图4所示[9],磁场区向外形变,电场区向内形变。Eacc为1.61 MV/m时,CST仿真计算的洛伦兹力分布如图5所示,腔颈处洛伦兹力最大,峰值为40 N/m2,赤道椭球面的力约10 N/m2。
根据式(2),在不同的Eacc下,洛伦兹力产生的腔体形变和频率偏移量不同,频率偏移量∆f与腔体加速梯度平方
$E_{{\rm{acc}}}^2$ 成正比关系[10-11]:$$\Delta f = {K_{\rm{L}}} \bullet E_{{\rm{acc}}}^2,$$ (5) 其中KL称为洛伦兹力失谐因子,是一个衡量腔在洛伦兹力下机械稳定性的参量,KL越小,说明超导腔越不易产生洛伦兹力失谐。
在两端束管端口固定边界条件下,用软件COMSOL Multiphysics计算超导椭球腔的洛伦兹力失谐量随Eacc的变化,如图6所示,拟合曲线计算KL为1.574 Hz/(MV/m)2,当设计工作加速梯度为10 MV/m时,腔体的洛伦兹失谐量为157.4 Hz。
Detuning Analysis of the Superconducting Elliptical Cavity for CSNS-II Linac
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摘要: 中国散裂中子源二期束流功率升级到500 kW,直线加速器H-的能量增益由现在的80 MeV提高到300 MeV以上,其中150 ~ 300 MeV能量段采用648 MHz βg=0.60的5-cell超导椭球腔结构。超导椭球腔具有加速梯度高、结构简单、后处理容易等优点,缺点是结构强度弱、易失谐。本文主要研究该椭球腔的失谐特性。利用COMSOL Multiphysics软件进行了计算分析,在两端固定边界条件下,裸腔的氦压敏感性系数KP=–45.705 Hz/mbar(1 mbar=100 Pa),洛伦兹力失谐因子KL=1.574 Hz/(MV/m)2。增加加强环并优化其位置来改善氦压敏感性系数和洛伦兹力失谐因子,通过计算分析最终确定选择双加强环方案来减小椭球腔的失谐。两个加强环位置分别取在75和120 mm的位置,腔体失谐的改善最大,KP=6 Hz/mbar,KL=0.43 Hz/(MV/m)2。为了更准确地计算椭球腔的氦压敏感性系数和洛伦兹力失谐因子,本文引入调谐刚度边界条件进行失谐分析,在椭球腔调谐器端设置30 kN/mm边界条件,另一端固定,计算得到氦压敏感性系数为4.8 Hz/mbar,洛伦兹力失谐因子1.99 Hz/(MV/m)2,满足工程要求。另外,用CST软件对椭球腔的动态洛伦兹力失谐进行了初步分析,用软件Workbench计算了椭球腔的振动本征频率,结果显示,振动本征频率离射频脉冲重复频率及环境振动频率较远,不易发生共振失谐。Abstract: China Spallation Neutron Source(CSNS) beam power will upgrade to 500 kW(CSNS-II), energy gain of H- linac will up to 300 MeV from 80 MeV, and the 648 MHz βg=0.6 5-cell superconducting elliptical cavity will be adopted to accelerate the H- from 150 to 300 MeV. Accompanied with the advantage of high accelerating gradient, simple structure, easy post-processing, weak stiffeness and detuning is the shortcoming of elliptical cavity, this study analysis detuning characteristic of the elliptical cavity. With code COMSOL Multiphysics, helium pressure sensitivity coefficient KP and lorentz force detuning factor KL of bare cavity was calculated under two end fixed conditions, KP=–45.705 Hz/mbar(1 mbar=100 Pa) and KL=1.574 Hz/(MV/m)2. Stiffener ring was designed and optimized to reduce the Helium pressure sensitivity coefficient and lorentz force detuning factor, and double ring was chosen to improve the detuning of elliptical cavity at last by vast calculation and analysis, the position of stiffener ring was optimized at 75 and 120 mm under two end fixed boundary, Helium pressure sensitivity coefficient drop down to KP=6 Hz/mbar, lorentz force detuning factor drop down to KL=0.43 Hz/(MV/m)2. Moreover, elastic boundary was used to analyze helium pressure sensitivity coefficient and lorentz force detuning factor, more realistic results were got under tuner end 30 kN/mm boundary and the other end fixed condition, KP=4.8 Hz/mbar and KL=1.99 Hz/(MV/m)2, meet engineering requirements. In addition, dynamic lorentz force detuning was analyzed qualitatively with code CST. And natural frequency of cavity was calculated by code Workbench, results show that nature frequency is far from RF pulse repetition rate and ambient vibration frequency, no resonance happen.
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表 1 铌材特性
参量 值 杨氏模量/GPa 123 泊松比 0.38 密度/(kg/m3) 8.57×103 壁厚/mm 3.8 -
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